Elect. Comm. in Probab. 9 (2004), 178–187
ELECTRONIC
COMMUNICATIONS
in PROBABILITY
THE CENTER OF MASS OF THE ISE AND THE WIENER
INDEX OF TREES
SVANTE JANSON
Departement of Mathematics, Uppsala University, P. O. Box 480, S-751 06 Uppsala, Sweden.
email: svante.janson@math.uu.se
PHILIPPE CHASSAING
Institut Élie Cartan, Université Henri Poincaré-Nancy I, BP 239, 54506 Vandœuvre-lèsNancy, France.
email: chassain@iecn.u-nancy.fr
Submitted 31 October 2003, accepted in final form 17 November 2004
AMS 2000 Subject classification: 60K35 (primary), 60J85 (secondary).
Keywords: ISE, Brownian snake, Brownian excursion, center of mass, Wiener index.
Abstract
We derive the distribution of the center of mass S of the integrated superBrownian excursion
(ISE) from the asymptotic distribution of the Wiener index for simple trees. Equivalently, this
is the distribution of the integral of a Brownian snake. A recursion formula for the moments
and asymptotics for moments and tail probabilities are derived.
1
Introduction
The Wiener index w(G) of a connected graph G with set of vertices V (G) is defined as
w(G) =
X
d(u, v),
(u,v)∈V (G)
in which d(u, v) denotes the distance between u and v in the graph.
The ISE (integrated superBrownian excursion) is a random variable, that we shall denote J ,
with value in the set of probability measures on Rd . The ISE was introduced by David Aldous
[1] as an universal limit object for random distributions of mass in Rd : for instance, Derbez
& Slade [11] proved that the ISE is the limit of lattice trees for d > 8.
A motivation for this paper comes from a tight relation established between some model of
random geometries called fluid lattices in Quantum geometry [3], or random quadrangulations
in combinatorics, and the one-dimensional ISE [6]: let (Qn , (bn , en )) denote the random uniform choice of a quadrangulation Qn with n faces, and of a marked oriented edge en with root
178
The center of mass of the ISE
179
vertex bn in Qn , and set
rn
=
Wn
=
max d(x, bn ),
X
d(x, bn ),
x∈V (Qn )
x∈V (Qn )
rn and¡Wn being the radius
of Qn 1 , respectively. As a consequence
¢ and the total path length1/4
−1/4
−5/4
of [6], n
rn , n
Wn converges weakly to (8/9) · (R − L, S − L), in which
Z
S = xJ (dx)
stands for the center of mass of the ISE, while, in the case d = 1, [L, R] denotes the support
of J . In [8], a modified Laplace transform is given, that determines the joint law of (R, L).
As a consequence, the first two moments of R − L are derived, and this is pretty much all the
information known about the joint law of (R, L, S), to our knowledge. In this paper, we prove
that
Theorem 1.1. For k ≥ 0,
£
¤
E S 2k =
√
(2k)! π
ak ,
2(9k−4)/2 Γ ((5k − 1)/2)
(1)
in which ak is defined by a1 = 1, and, for k ≥ 2,
ak = 2(5k − 4)(5k − 6)ak−1 +
k−1
X
ai ak−i .
(2)
i=1
Since S has a symmetric distribution, the odd moments vanish. Also, we have the following
asymptotics for the moments.
Theorem 1.2. For some constant β = 0.981038 . . . we have, as k → ∞,
¡
¢−2k/4
£
¤ 2π 3/2 β
3
(2k)1/2 10e3
(2k) 4 ·2k .
(3)
E S 2k ∼
5
As a consequence, Carleman’s condition holds, and the distribution of S is uniquely determined
by its moments.
In Section 2, we recall the description of J in terms of the Brownian snake, following [14, Ch.
IV.6], and we derive a distributional identity for S in term of some statistic
Z
min e(u) ds dt.
(4)
η=4
0≤s<t≤1 s≤u≤t
of the normalized Brownian excursion (e(t))0≤t≤1 . In [12], the joint moments of (η, ξ), in
which
Z 1
ξ=2
e(t) dt,
(5)
0
are computed with the help of explicit formulas for the joint moments of the Wiener index
and the total path length of random binary trees. In Section 3, we use the results of [12] to
derive Theorems 1.1 and 1.2. As a byproduct we also obtain that
1 incidentally,
nE [Wn | Qn ] /2 is the Wiener index of Qn .
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Theorem 1.3. We have, as k → ∞,
£ ¤ 2π 3/2 β 1/2
−k/2 k/2
k (5e)
k ,
E ηk ∼
5
(6)
Finally, Section 4 deals with tail estimates for S.
2
The ISE and the Brownian snake
For the ease of non specialists, we recall briefly the description of J in terms of the Brownian
snake, from [14, Ch. IV.6] (see also [5, 10, 16], and the survey [17]). The ISE can be seen as
the limit of a suitably renormalized spatial branching process (cf. [6, 15]), or equivalently, as
an embedding of the continuum random tree (CRT) in Rd .
As for the Brownian snake, it can be seen as a description of a ”continuous” population
T , through its genealogical tree and the positions of its members. Given the lifetime process
ζ = (ζ(s))s∈T of the Brownian snake, a stochastic process with values in [0, +∞), the Brownian
snake with lifetime ζ is a family
W = (Ws (t))0≤s≤1,
0≤t≤ζ(s)
of stochastic processes Ws (·) with respective lifetimes ζ(s). The lifetime ζ specifically describes
the genealogical tree of the population T , and W describes the spatial motions of the members
of T . A member of the population is encoded by the time s it is visited by the contour traversal
of the genealogical tree, ζ(s) being the height of member s ∈ T in the genealogical tree (ζ(t)
can be seen as the ”generation” t belongs to, or the time when t is living). Let
C(s, t) = min ζ(u),
s≤u≤t
s ∧ t = argmin ζ(u).
s≤u≤t
Due to the properties of the contour traversal of a tree, any element of s ∧ t is a label for the
more recent ancestor common to s and t, and the distance between s and t in the genealogical
tree is
d(s, t) = ζ(s) + ζ(t) − 2C(s, t).
If it is not a leaf of the tree, a member of the population is visited several times (k + 1 times
if it has k sons), so it has several labels: s and t are two labels of the same member of the
population if d(s, t) = 0, or equivalently if s ∧ t ⊃ {s, t}. Finally, s is an ancestor of t iff
s ∈ s ∧ t. In this interpretation, Ws (u) is the position of the ancestor of s living at time u, and
cs = Ws (ζ(s)) ,
W
s ∈ T,
is the position of s. Before time m = C(s1 , s2 ), s1 and s2 share the same ancestor, entailing
that
(7)
(Ws1 (t))0≤t≤m = (Ws2 (t))0≤t≤m .
Obviously there is some redundancy in³ this description:
it turns out that the full Brownian
´
c
snake can be recovered from the pair Ws , ζ(s)
(see [15] for a complete discussion of
0≤s≤1
this).
In the general setting [14, Ch. IV], the spatial motion of a member of the population is any
Markov process with cadlag paths. In the special case of the ISE, this spatial motion is a
d-dimensional Brownian motion:
The center of mass of the ISE
181
a) for all 0 ≤ s ≤ 1, t → Ws (t) is a standard linear Brownian motion started at 0, defined
for 0 ≤ t ≤ ζ(s) ;
b) conditionally, given ζ, the application s → Ws (.) is a path-valued Markov process
with transition function defined as follows: for s1 < s2 , conditionally given Ws1 (.),
(Ws2 (m + t))0≤t≤ζ(s2 )−m is a standard Brownian motion starting from Ws1 (m), independent of Ws1 (.).
The lifetime ζ is usually a reflected linear Brownian motion [14], defined on T = [0, +∞).
However, in the case of the ISE,
ζ = 2e,
in which e denotes the normalized Brownian excursion, or 3-dimensional Bessel bridge, defined
on T = [0, 1]. With this choice of ζ, the genealogical tree is the CRT (see [1]), and the Brownian
snake can be seen as an embedding of the CRT in Rd . We can now give the definition of the
ISE in terms of the Brownian snake with lifetime 2e [14, Ch. IV.6]:
c.
Definition 2.1. The ISE J is the occupation measure of W
c is defined by the relation:
Recall that the occupation measure J of a process W
Z 1 ³ ´
Z
cs ds,
f (x)J (dx) =
f W
R
(8)
0
holding for any measurable test function f .
Remark 2.2. It might seem more natural to consider the Brownian snake with lifetime e
instead of 2e, but we follow the normalization in Aldous [1].
√ If we used the Brownian snake
with lifetime e instead, W , J and S would be scaled by 1/ 2.
Based on the short account on the Brownian snake given in this Section, we obtain:
Theorem 2.3. Let N be a standard Gaussian random variable, independent of η. Then
law √
S = η N.
Proof. Specializing (8) to f (x) = x, we obtain a representation of S:
Z 1
cs ds,
S=
W
0
which is the starting point of our proof. We have also, directly from the definition of the
Brownian snake,
³ ´
cs
Proposition 2.4. Conditionally, given e, W
is a Gaussian process whose covariance
0≤s≤1
is C(s, t) = 2 mins≤u≤t e(u), s ≤ t.
Proof. With the notation ζ = 2e and m = C(s1 , s2 ) = 2 mins1 ≤u≤s2 e(u), we have, conditionally, given e, for s1 ≤ s2 ,
´
³
cs
cs , W
= Cov (Ws1 (ζ(s1 )), Ws2 (ζ(s2 )) − Ws2 (m) + Ws2 (m))
Cov W
2
1
=
=
Cov (Ws1 (ζ(s1 )), Ws2 (m))
Cov (Ws1 (ζ(s1 )), Ws1 (m))
=
m,
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in which b) yields the second equality, (7) yields the third one, and a) yields the fourth
equality.
As a consequence of Proposition 2.4, conditionally given e, S is centered Gaussian with variance
Z
C(s, t) ds dt = η.
[0,1]2
This last statement is equivalent to Theorem 2.3.
Remark 2.5. The d-dimensional analog of Theorem 2.3 is also true, by the same proof.
3
The moments
Proof of Theorem 1.1. From Theorem 2.3, we derive at once that
£
¤
£ ¤ £
¤
E S 2k = E η k E N 2k .
∗
As a special case of [12, Theorem 3.3] (where ak is denoted ω0k
),
√
£ ¤
k! π
ak ,
E η k = (7k−4)/2
2
Γ ((5k − 1)/2)
£
¤
and the result follows since E N 2k = (2k)!/(2k k!).
(9)
In particular, see again [12, Theorem 3.3],
p
£ ¤
E S 2 = E [η] = π/8,
£ ¤
£ ¤
E S 4 = 3E η 2 = 7/5,
as computed by Aldous [1] by a different method. (Aldous’ method extends to higher moments
too, but the calculations quickly become complicated.) The asymptotics for the moments are
obtained as follows.
Proof of Theorems 1.2 and 1.3. We just prove that
ak ∼ β 50k−1 (k − 1)!2 ,
Set
bk =
(10)
ak
.
50k−1 (k − 1)!2
We have b1 = a1 = 1 and, from (2), b2 = b3 = 49/50 = 0.98. For k ≥ 4, (2) translates to
bk
=
=
Pk−1
¶
1
i=1 ai ak−i
bk−1 + k−1
2
25(k − 1)
50
(k − 1)!2
Pk−2
i=2 ai ak−i
bk−1 + k−1
,
50
(k − 1)!2
µ
1−
Thus bk increases for k ≥ 3. We will show that bk < 1 for all k > 1; thus β = limk→∞ bk exists
and (10) follows. Stirling’s formula and (1), (9) then yield (6) and (3).
The center of mass of the ISE
183
More precisely, we show by induction that for k ≥ 3,
bk ≤ 1 −
1
.
25(k − 1)
This holds for k = 3 and k = 4. For k ≥ 5 we have by the induction assumption b j ≤ 1 for
1 ≤ j < k and thus
sk =
Pk−2
i=2 ai ak−i
k−1
50
(k − 1)!2
k−2
1 X (i − 1)!2 (k − i − 1)!2
bi bk−i
50 i=2
(k − 1)!2
Ã
!
1
4
≤
2 + (k − 5)
2
2
2
50 (k − 1) (k − 2)
(k − 3)
1
.
≤
2
25 (k − 1) (k − 2) (k − 3)
=
(11)
1
The induction follows. Moreover, it follows easily from (11) that bk < β < bk + 75
(k − 2)−3 .
To obtain the numerical value of β, we write, somewhat more sharply, where 0 ≤ θ ≤ 1,
sk =
1
2
25 (k − 1) (k − 2)
2 b2 bk−2
+
4
2
2
25 (k − 1) (k − 2) (k − 3)2
+ θ(k − 7)
b3 bk−3
36
2
2
25 (k − 1) (k − 2) (k − 3)2 (k − 4)2
and sum over k > n for n = 10, say, using bn−1 < bk−2 < β and bn−2 < bk−3 < β for k > n.
It follows (with Maple) by this and exact computation of b1 , . . . , b10 that 0.981038 < β <
0.9810385; we omit the details.
Remark 3.1. For comparison, we give the corresponding result for ξ defined in (5). There is a
simple relation, discovered by Spencer [18] and Aldous [2], between its moments and Wright’s
constants in the enumeration of connected graphs with n vertices and n + k edges [19], and
the well-known asymptotics of the latter lead, see [12, Theorem 3.3 and (3.8)], to
√
£ ¤
E ξ k ∼ 3 2k(3e)−k/2 k k/2 .
(12)
4
Moment generating functions and tail estimates
£ ¤
The moment asymptotics yield asymptotics for the moment generating function E etS and
the tail probabilities P (S > t) as t → ∞. For completeness and comparison, we also include
corresponding results for η. We begin with a standard estimate.
Lemma 4.1.
(i) If γ > 0 and b ∈ R, then, as x → ∞,
∞
X
k=1
k b k −γk xk ∼
³ 2π ´1/2 ¡
γ
e−γ x
¢(b+1/2)/γ
eγ(e
−γ
x)1/γ
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Electronic Communications in Probability
(ii) If −∞ < γ < 1 and b ∈ R, then, as x → ∞,
∞
X
k b k γk xk
k=1
k!
¡
¢b/(1−γ) (1−γ)(eγ x)1/(1−γ)
∼ (1 − γ)−1/2 eγ x
e
The sums over even k only are asymptotic to half the full sums.
Sketch of proof. (i). This is standard, but since we have not found a precise reference, we
sketch the argument. Write k b k −γk xk = ef (k) = k b eg(k) where g(y) = −γy ln y + y ln x and
f (y) = g(y) + b ln y. The function g is concave with a maximum at y0 = y0 (x) = e−1 x1/γ . A
Taylor expansion yields
Z ∞
∞
X
−1/2
−1/2 −f (y0 )
b −γk k
ef (dye)−f (y0 ) dy
k k
x = y0
y0
e
k=1
=
→
Z
0
∞
1/2
1/2
−y0
Z
ef (dy0 +sy0
∞
e−γs
2
/2
ds =
−∞
e)−f (y0 )
ds
³ 2π ´1/2
γ
,
in which dye denotes the smallest integer larger than, or equal to, y.
(ii). Follows by (i) and Stirling’s formula.
tη
Combining
Lemma 4.1,
£ k ¤ kTheorem 1.2
£ tSand
¤ P
£ k ¤wek find the following asymptotics for E [e ] =
P
t /k! and E e
= k even E S t /k!.
kE η
Theorem 4.2. As t → ∞,
£ ¤ (2π)3/2 β t2 /10
te
,
E etη ∼
53/2
£ ¤ 21/2 π 3/2 β 2 t4 /40
E etS ∼
t e
.
53/2
(13)
(14)
Proof. For η we take b = 1/2, γ = 1/2, x = (5e)−1/2 t in Lemma 4.1(ii); for S we take b = 1/2,
γ = 3/4, x = (10e3 )−1/4 t.
The standard argument with Markov’s inequality yields upper bounds for the tail probabilities
from Theorem 1.2 or 4.2.
Theorem 4.3. For some constants K1 and K2 and all x ≥ 1, say,
¡
¢
(15)
P (η > x) ≤ K1 x exp − 25 x2 ,
¡ 3 1/3 4/3 ¢
2/3
.
(16)
P (S > x) ≤ K2 x exp − 4 10 x
£
¤
Proof. For any even k and x > 0, P (|S| > x) ≤ x−k E S k . We use (3) and optimize the
resulting exponent by choosing k = 101/3 x4/3 , rounded to an even integer. This yields (16);
we omit the details. (15) is obtained similarly from (6), using k = b5x2 c.
Remark 4.4. The proof of Theorem 4.3 shows that any K1 > 2π 3/2 β/51/2 ≈ 4.9 and
K2 > 101/6 βπ 3/2 /5£ ≈ ¤1.6 will do for large x. Alternatively, we could use Theorem 4.2 and
P (S > x) < e−tx E etS for t > 0 and so on; this would yield another proof of Theorem 4.3
with somewhat inferior values of K1 and K2 .
The center of mass of the ISE
185
The bounds obtained in Theorem 4.3 are sharp up to factors 1 + o(1) in the exponent, as
is usually the case for estimates derived by this method. For convenience we state a general
theorem, reformulating results by Davies [7] and Kasahara [13].
Theorem 4.5. Let X be a random variable, let p > 0 and let a and b be positive real numbers
related by a = 1/(pebp ) or, equivalently, b = (pea)−1/p .
(i) If X ≥ 0 a.s., then
is equivalent to
− ln P(X > x) ∼ axp
¡
E Xr
¢1/r
∼ br1/p
as x → ∞
as r → ∞.
(17)
(18)
Here r runs through all positive reals; equivalently, we can restrict r in (18) to integers
or even integers.
(ii) If X is a symmetric random variable, then (17) and (18) are equivalent, where r in (18)
runs through even integers.
(iii) If p > 1, then, for any X, (17) is equivalent to
¢
¡
as t → ∞,
ln E etX ∼ ctq
(19)
where 1/p + 1/q = 1 and c = q −1 (pa)−(q−1) = q −1 eq−1 bq . (This can also be written in
the more symmetric form (pa)q (qc)p = 1.) Hence, if X ≥ 0 a.s., or if X is symmetric
and r restricted to even integers, (19) is also equivalent to (18).
Proof. For X ≥ 0, (i) and (iii) are immediate special cases of more general results by Kasahara [13, Theorem 4 and Theorem 2, Corollary 1], see also [4, Theorem 4.12.7]; the difficult
implications (18) =⇒ (17) and (19) =⇒ (17) were earlier proved by Davies [7] (for p > 1,
which implies the general case of (i) by considering a power of X). Moreover, (19) =⇒ (17)
follows also from the Gärtner–Ellis theorem [9, Theorem 2.3.6] applied to n −1/p X. Note that,
assuming X ≥ 0, (E X r )1/r is increasing in r > 0, which implies that (18) for (even) integers
is equivalent to (18) for all real r.
(ii) follows from (i) applied to |X|, and (iii) for general X follows by considering max(X, 0).
We thus obtain from Theorem 1.2 or 4.2 the following estimates, less precise than Theorem 4.3
but including both upper and lower bounds.
Theorem 4.6. As x → ∞,
¡
¢
5
ln P (η > x) ∼ − x2 ,
2
¡
¢
3
ln P (S > x) ∼ − 101/3 x4/3 .
4
(20)
(21)
Proof. For η we use Theorem 4.5 with p = q = 2, b = (5e)−1/2 , a = 5/2 and c = 1/10. For S
we have p = 4/3, q = 4, b = (10e3 )−1/4 , a = 3 · 101/3 /4 and c = 1/40.
Remark 4.7. (20) can also be proved using the representation (4) and large deviation theory
for Brownian excursions, cf. [9, §5.2] and [10]. The details may perhaps appear elsewhere.
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Electronic Communications in Probability
Remark 4.8. (21) can be compared to the tail estimates in [1] for the density function of
cU , the value of the Brownian snake at a random point U , which in particular gives
W
´¢
¡ ³
cU > x ∼ −3 · 2−5/3 x4/3 .
ln P W
Remark 4.9. For ξ we can by (12) use Theorem 4.5 with p = q = 2, b = (3e)−1/2 , a = 3/2
and c = 1/6. In [12], the variable of main interest is neither ξ nor η but ζ = ξ − η. By
£ ¤1/k
£ ¤1/k
£ ¤1/k
Minkowski’s inequality E ξ k
≤ E ζk
+ E ηk
and (6), (12) follows
¡ £ k ¤¢1/k
¡ £ k ¤¢1/k
E ζ
E ζ
1
1
1
√ − √ ≤ lim inf
≤ lim sup
≤√ .
1/2
1/2
k→∞
k
k
3e
5e
3e
k→∞
This leads to asymptotic upper and lower bounds for ln(P (ζ > x)) too by [7] or [13]. We
¡ £ ¤¢1/k
can show that limk→∞ k −1/2 E ζ k
and limx→∞ x−2 ln(P (ζ > x)) exist, but do not know
their value.
Acknowledgement. We thank Jim Fill and anonymous referees for helpful comments.
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The center of mass of the ISE
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